Application of Derivative ASSIGNMENT FOR IIT-JEE

7/29/2019 Application of Derivative ASSIGNMENT FOR IIT-JEE

1/15

1. The normal to the curve )cos(sinay),sin(cosax =+= at any point '' is such that

(a) It passes thorugh

a,

2a (b) It is at a constant distance from the origin

(c) It passes through the origin (d) It makes an angle +2

with the x-axis

2. The tangents to the curve y = e2x at the point (0, 1) meets the x-axis at

(a) (0, 0) (b) (2, 0) (c)

0,

2

1(d) none of these

3. Tangents to the curve y = x3 at x = -1 and x = 1 are(a) parallel (b) intrersecting obliquely but not at an angle of 450

(c) perpendicular to each other (d) intersecting at an angle of 450

4. The slope of the normal to the curve x3 = 8a2y, a > 0, at a point in the first quadrant is3

2 , then the point is

(a) (2a, -a) (b) (2a, a) (c) (a, 2a) (d) (a, a)

5. The point on the curve y = (x - 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) is

(a)

4

1,

2

7(b)

4

1,

2

5(c)

4

1,

2

5(d)

4

1,

2

7

6. The normal at the point (1, 1) on the curve 2y = 3 - x2 is(a) x + y = 0 (b) x + y + 1 = 0 (c) x - y + 1 = 0 (d) x - y = 0

7. For the curve == 0,sinex,cossin3y ; the tangent is parallel to x-axis when is

(a) 0 (b)2

(c)

4

(d)

6

8. If the rate of increase of 5x22

x 2+ is twice the rate of decrease of it, then x is

(a) 2 (b) 3 (c) 4 (d) 1

9. The curve 2b

y

a

xnn

=

+

touches the straight line 2

b

y

a

x=+ at the point (a, b) for

(a) n = 3 (b) n = 2 (c) any value of n (d) no value of n

10. The point on the curve y = 6x - x2 where the tangent is parallel to x-axis is(a) (0, 0) (b) (2, 8) (c) (6, 0) (d) (3, 9)

LEVEL - 1 (Objective)

Application ofDarivatives


2/15

11. If the curve y = ax2 - 6x + b passes through (0, 2) and has its tangent parallel to x-axis at2

3x = , then the values

of a and b are respectively(a) 2, 2 (b) -2, -2 (c) -2, 2 (d) 2, -2

12. For the curve x = t2 - 1, y = t2 - t, the tangent line is perpendicular to x-axis, where

(a) t = 0 (b) t (c) 31

t = (d) 31

t =

13. The line 1b

y

a

x=+ touches the curve y = be-x/a at the point

(a)

a

b,a (b)

a

b,a (c)

b

a,a (d) none of these

14. The slope of tangent to the curve x = t2 + 3t - 8, y = 2t2 - 2t - 5 at the point (2, -1) is

(a) 722

(b) 76

(c) -6 (d) none of these

15. Equation of the tangent at the point P(t), where t is any parameter, to the parabola y2 = 4ax is

(a) yt = x + at2 (b) y = xt + at2 (c) y = tx (d)t

atxy +=

16. The point on the curve y2 = x, the tangent at which makes an angle of 450 with x-axis will be given by

(a)

4

1,

2

1(b)

2

1,

2

1(c) (2, 4) (d)

2

1,

4

1

17. The straight line x + y = a will be a tangent to the ellipse 116

y

9

x 22=+ if 'a' =

(a) 8 (b) 5 (c) 10 (d) 618. The equation to the normal to the curve y = sin x at (0, 0) is

(a) x = 0 (b) y = 0 (c) x + y = 0 (d) x - y = 019. If tangent to the curve x = at2, y = 2at is perpendicular to x-axis then its point of contact is

(a) (a, a) (b) (0, a) (c) (a, 0) (d) (0, 0)20. Equation of tangent to the curve x = a cos3t , y = a sin3t at 't' is

(a) x sec t - y cosec t = a (b) x sec t + y cosec t = a (c) x cosec t + y cosec t = a (d) none of these21. The angle of intersection of the curves y = x2 and 6y = 7 - x3 at (1, 1) is

(a)4

(b)

3

(c)

2

(d) none of these

22. The angle of intersection of the curves y = 4 - x2 and y = x2 is

(a)2

(b)

3

4tan

1 (c)7

24tan

1 (d) none of these

23. Each curve of the system x2 - y2 = p cuts each curve of the system xy = q at an angle

(a)6 (b)

4 (c)

3 (d)

2


3/15

24. Two curves x3 - 3xy2 + 2 = 0 and 3x2y - y3 - 2 = 0(a) cut at right angle (b) touch each other

(c) cut at an angle3

(d) cut at an angle

4

25. The normal at the point (bt12, 2bt

1) on a parabola meets the parabola again in the point (bt

22, 2bt

2) then

(a)1

12t2tt += (b)

1

12t2tt = (c)

1

12t2tt += (d)

1

12t2tt =

26. A point on the parabola y2 = 18x at which the ordinate increases at twice the rate of the abscissa is

(a) (2, 4) (b) (2, -4) (c)

2

9,

8

9(d)

2

9,

8

9

27. A function y = f(x) has a second order derivative )1x(6)x(f = . If its graph passes through the point (2, 1)

and at that point the tangent to the graph is y = 3x - 5, then the function is(a) (x - 1)2 (b) (x - 1)3 (c) (x + 1)3 (d) (x + 1)2

28. The normal to the curve =+= sinay),cos1(ax at '0' always passes through the fixed point(a) (a, 0) (b) (0, a) (c) (0, 0) (d) (a, a)

29. If 2a + 3b + 6c = 0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval(a) (0, 1) (b) (1, 2) (c) (2, 3) (d) (1, 3)

30. Let f be differentiable for all x. If f(1) = -2 and 2)x(f for ]6,1[x , then

(a) f(1) < 5 (b) f(6) = 5 (c) 8)6(f (d) f(6) < 8

31. If the function f(x) = ax3 + bx2 + 11x - 6 satisfies condition of Rolle's theorem in [1, 3] and 03

12f =

+ , then

values of 'a' and 'b' are respectively

(a) 1, -6 (b) -2, 1 (c)2

1,1 (d) -1, 6

32. Rolle's theorem holds for the function x3 + bx2 + cx, 2x1 at the point3

4, the value of b and c are:

(a) b = 8, c = -5 (b) b = -5, c = 8 (c) b = 5, c = -8 (d) b = -5, c = -8

33. Let f(x) and g(x) be differentiable for 1x0 such that f(0) = 0, g(0) = 0, f(1) = 6. Let there exist a real

number 'c' in (0, 1) such that )c(g2)c(f = , then the value of g(1) must be(a) 1 (b) 3 (c) -2 (d) -1

34. If the function f(x) = x3 - 6x2 + ax + b defined on [1, 3] satisfies Rolle's theorem for

+=3

12c , then

(a) a = 11, b = 6 (b) a = -11, b = 6 (c) a = 11, Rb (d) none of these

35. If 'f ' is strictly increasing function, then =

)0(f)x(f

)x(f)x(flim

2

0x

(a) 0 (b) 1 (c) -1 (d) 2


4/15

36. For the function f(x) = x2 - 6x + 8, ,4x2 the value of 'x' for which )x(f vanishes is

(a) 3 (b)2

5(c)

4

9(d)

2

7

37. The function f(x) = x3 - 3x is

(a) increasing in ),1()1,( and decreasing in (-1, 1)

(b) decreasing in ),1()1,( and increasing in (-1, 1)(c) increasing in ),0( and decreasing in )0,(

(d) decreasing in ),0( and increasing in )0,(

38. The function y = x3 + 5x2 - 1 is decreasing in the interval

(a) 0x3

10


5/15

49. The point of the hyperbola 118

y

24

x22

= which is nearest to the line 3x + 2y + 1 = 0 is

(a) (6, -3) (b) (-6, 3) (c) (3, 6) (d) (6, 3)

50. Given: f(x) = x1/x, (x > 0) has the maximum value at x = e, then

(a) ee > (b) ee > (c) ee = (e) ee


6/15

1. For a certain curved y

dx

2

2 = 6x - 4 and y has a local minimum value 5 when x = 1 .

Find the equation and the global maximum and minimum values of y, given that 0 x 2 .2. Suppose f(x) is real valued polynomial function of degree 6 satisfying the following conditions ;

(a) f has minimum value at x = 0 and 2(b) f has maximum value at x = 1

(c) for all x, Limitx 0

1x

lnf xx

x

x

( ) 1 0

0 1 1

1 0 1

= 2 . Determine f(x).

3. If x > 0, let f(x) = 5x2 + Ax -5 , where A is a positive constant . Find the smallest A such that f(x) 24 for allx > 0 .

4. Let xsinxsin)x(f 23 += where2

x2

+= is a constant

6. A cubic f(x) vanishes at x = -2 and has a relative min./max. at x = -1 and x = 1/3. If

=1

1

,3/14dx)x(f then find

the cubic function f(x)

7. Let


7/15

13. Determine the condition so that the function f(x) = x3 + px2 + qx + r is an increasing function for all realx.

14. Letx

1

ln tf (x) dt, x R f (x) and g(x) f (x) f (1/ x) x R .

1 t

+ += = + + Find )(xg in terms of x and discuss its

monotonicity.

15. Let )2,0(x0)x(fand)x2(f2

xf2)x(g + = 0x,xaxx

0x,xe)x(f32

ax

where a is a positive constant. Then find the interval in

which )x(f is increasing.19. Show that the curve y = bex/a, the subtangent is of constant length and subnormal varies as the square of

ordinate?20. Find the equation of tangent and normal, the length of subtangent and subnormal of the circle x2 + y2 = a2 at the

point (x1, y

1).

21. Find the equation of normal to the curve x + y = xy, where it cuts x-axis.22. Let P be any point on the curve x2/3 + y2/3 = a2/3. Then find the length of the segment of the tangent between the

coordinate axes.23. If the relation between sub-normal SN and subtangent ST at any point S on the curve; by2 = (x + a)3 is

p(SN) = q(ST)2, then find the value ofq

p.

24. Find the equation of normal to the curve y = (1 + x)y + sin-1(sin2 x) at x = 0.

25. Find all tangents to the curve y = cos (x + y), 2x2 that are parallel to the line x + 2y = 0.26. Determine the parameters a, b, c in the equation of the curve y = ax2 + bx + c so that y = x is a tangent to the

curve at x = 1 and the curve passes through the point (-1, 0).

27. If two variables x and y are such that x > 0 and xy = 1, find the minimum value of x + y.28. Find the maximum slope of the curve y = -x3 + 3x2 + 9x - 27 and at what point is it?

29. Show that the curves ax2 + by2 = 1 and a1x2 + b

1y2 = 1 cut each other orthogonally if

b

1

b

1

a

1

a

1

11

= .

30. Use the function1

xf (x) x , x 0= > to show that ee > .

31. Show that1x

x

11)x(f

+

+= is a decreasing function for x > 0.

32. Show that 21 x1xlogxtanx2 ++ for x > 0.


8/15

1. The minimum value of the function y = 2x3 - 21x2 + 36x - 20 is(a) -128 (b) -126 (c) -120 (d) none of these

2. The maximum value of the function sin x (1 + cos x) is

(a) 3 (b)4

33(c) 4 (d) 33

3. f(x) = sin x + cos 2x (x > 0) has minima for x =

(a) 2n (b) +3 )1n(2 (c) +

)1n2(21 (d) none of these

4. The difference between the greatest and the least value of dt)t1(e)x(fx

0

t += in the interval [1, 2] is

(a) 2e2 - e (b) 3e2 - 2e (c) 2e (d) none of these5. If the function f(x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0, attains its maximum and minimum at p and q

respectively such that p2 = q, then a equals

(a) 1 (b) 2 (c) 2

1

(d) 3

6. The real number x when added to its inverse gives the minimum value of the sum at x equal to(a) 1 (b) -1 (c) -2 (d) 2

7. The length of subnormal to the parabola y2 = 4ax at any point is equal to

(a) a2 (b) a22 (c)2

a(d) 2a

8. Sum of squares of intercepts made on coordinate axes by the tangent to the curve x2/3 + y2/3 = a2/3 is(a) a2 (b) 2a2

(c) 3a2 (d) 4a29. The length of normal at 't' on the curve x = a(t + sin t), y = a(1 - cos t) is

(a) a sin t (b)2

tsec

2

tsina2

3

(c)2

ttan

2

tsina2 (d)

2

tsina2

10. The length of the subtangent to the curve x2 + xy + y2 = 7 at (1, -3) is

(a) 3 (b) 5 (c) 15 (d)5

3

11. For the parabola y2 = 4ax, the ratio of the subtangent to the abscissa is

(a) 1 : 1 (b) 2 : 1(c) x : y (d) x2 : y

LEVEL - 3(Questions asked from previous Engineering Exams)


9/15

12. The abscissae of the points where the tangent to curve y = x3 - 3x2 - 9x + 5 is parallel to x-axis are(a) x = -1, 3 (b) x = -3, 1(c) x = 1, -1 (d) x = 0

13. The equation of the tangent ot eh curve y = be-x/a at the point where it crosses the y-axis is

(a) 1b

y

a

x= (b) ax + by = 1 (c) ax - by = 1 (d) 1

b

y

a

x=+

14. If a < 0, the function f(x) = eax + e-ax is monotonically decreasing for all values of x, where(a) x < 0 (b) x > 0 (c) x < 1 (d) x > 1

15. The maximum value ofx

xlogin ),2[ is

(a)2

2log(b) 0 (c)

e

1(d) 1

16. For what value of x, the function x3 + 3x2 + 3x + 7 is increasing(a) for all x (b) for x < 0 (c) for x > 0 (d) for x = 0

17. Let x, y be two variables and x > 0, xy = 1, then minimum value of x + y is

(a) 1 (b) 2 (c)2

12 (d)

3

13

18. If f(x) = ex, ]1,0[x , then a number 'c' of Lagrange's mean value theorem is

(a) log (e - 1) (b) log (e + 1) (c) log e (d) none of these19. The equation of the tangent to the curve (1 + x2)y = 2 - x where it crosses X-axis is

(a) x + 5y = 2 (b) x - 5y = 2 (c) 5x - y = 2 (d) 5x + y - 2 = 020. The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases,

when the side is 10 cm is

(a) 3 sq. units/sec (b) 10 sq. units/sec (c) 310 sq. units/sec (d)3

10sq. units/sec

21. Angle between y2 = x and x2 = y at origin is

(a) 2 tan-1(3/4) (b) tan-1 (4/3) (c)2

(d)

4

22. If x + y = 60; x, y > 0 then maximum value of xy3 is(a) 30 (b) 60 (c) 15(45)3 (d) 45(15)3

23. On [1, e], the greatest value of x2 log x is

(a) e2 (b)

e

1log

e

1(c) eloge2 (d) none of these

24. The function f(x) = x1/x is

(a) increasing in ),1(

(b) decreasing in ),1(

(c) increasing in (1, e) and decreasing in ),e(

(d) decreasing in (1, e) and increasing in ),e(


10/15

25. If ,0x),x1log(1x

1)x(f >+

+= then f is

(a) an increasing function (b) a decreasing function(c) both increasing and decreasing (d) none of these

26. The point on the curve ayx =+ , the normal at which is parallel to the x-axis is

(a) (0, 0) (b) (0, a) (c) (a, 0) (d) (a, a)27. If the line ax + by + c = 0 is a normal to the curve xy = 1, then(a) a > 0, b > 0 (b) a > 0, b < 0 (c) a < 0, b < 0 (d) none of these

28. The normal to the curve )cos(sinay),sin(cosax =+= at any is such that(a) it makes a constant angle with x-axis (b) it passes through origin(c) it is at a constant distance from the origin (d) none of these

29. The triangle formed by the tangent to the curve f(x) = x2 + bx - b at the point (1, 1) and the coordinate axes liesin the first quadrant. If its area is 2, then the value of b is(a) -1 (b) 3 (c) -3 (d) 1

30. If the parametric equation of a curve is given by x = et

cos t, y = et

sin t, then the tangent to the curve at the point

4t

= makes with the axis of x-axis the angle

(a) 0 (b) 4/ (c) 3/ (d) 2/31. The curve y - exy + x = 0 has a vertical tangent at the point

(a) (1, 1) (b) at no point (c) (0, 1) (d) (1, 0)

32. The equation of the tangent to the curve y = 2 sin x + sin 2x at3

x

= is equal to

(a) 33y2 = (b) 33y = (c) 033y2 =+ (d) 033y =+

33. The points on the curve2x1

xy

= where the tangent is inclined at angle

4

to x-axis are

(a) )2/3,3(),0,0( (b) )2/3,3(),0,0(

(c) )2/3,3(),0,0( (d) none of these

34. The coordiantes of the point on the curve y = x2 + 3x + 4 the tangent at which passes through the origin is equal to(a) (2, 14), (-2, 2) (b) (2, 14), (-2, -2)

(c) (2, 14), (2, 2) (d) none of these35. If y = 4x - 5 is a tangent to the curve y2 = ax3 + b at (2, 3), then

(a) a = 2, b = -7 (b) a = -2, b = 7(c) a = -2, b = -7 (d) a = 2, b = 7

36. If the normal to the curve y = f(x) at the point (3, 4) makes an angle4

3with the positive x-axis, then = )3(f

(a) -1 (b)4

3 (c)

3

4(d) 1

37. If x + y = k is normal to y2 = 12x, then k is(a) 3 (b) 9 (c) -9 (d) -3


11/15

38. The points on the curve y3 + 3x2 = 12y where the tangent is vertical, is (are)

(a)

2,

3

4(b)

1,

3(c) (0, 0) (d)

2,

3

4

39. Let

=

>=

0x,0

0x,xlnx)x(f Rolle's theorem is applicable to f for = if],1,0[x

(a) -2 (b) -1 (c) 0 (d)2

1

40. If f(x) = xex(1 - x), then f(x) is

(a) increasing on

1,

2

1(b) decreasing on R (c) increasing on R (d) decreasing on

1,

2

1

41. Let h(x) = f(x) - (f(x))2 + (f(x))3 for every real x, then(a) h is increasing whenever f is increasing (b) h is increasing whenever f is decreasing(c) h is decreasing whenever f is increasing (d) nothing can be said in general

42. The function f(x) = sin4x + cos4x increases if

(a)8

x0


12/15

50. If A > 0, B > 0 and3

BA

=+ , then the maximum value of tan A . tan B is

(a)3

1(b)

3

1

(c) 3 (d) 3

51. Let

1

f(x)min

when4

1bbx

2 += and b > 1

f(x) neither max nor min when b = 1.6. (x + 2)(x2 - x + 1) = 07. (-2, -1) [1, )

8. Equation of tangent x + 2y + 3a = 0Equation of normal 2x - y - 3a = 0

9.8

9,

8

9are its two max. values and 0, -2 are

its two min. values.10.

11. 105and21 +

12. ( ) +, ( )3 3 ( )3 1 ,13. p2 - 3q < 0.

14. 2)(ln2

1)( xxg = and its increasing in ),1(

and decreasing in (0, 1)

15. g(x) is increasing in

3

4,0 and decreasing

in

2,

3

4

LEVEL -2 (Subjective)

16. ),1(2

213,4a

17.

>

2

,4

xif0(x)g and


15/15

1. a

2. b

3. c

4. a

5. b6. a

7. d

8. a

9. c

10. c

11. b

12. a

13. d

14. a

15. c

16. a

17. b

18. a

19. a

20. c21. c

22. c

23. a

24. c

25. b

26. b

27. b

28. c

29. c

30. d

31. d

32. a33. a

34. a

35. a

36. d

37. b

38. d

39. d

40. a

41. a

42. b

43. b

44. c

45. a

46. d

47. b48. b

49. d

50. b

51. d

52. d

53. d

LEVEL - 3 (Questions asked from previous Engineering Exams)

ANSWER KEY

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Application of Derivative ASSIGNMENT FOR IIT-JEE